# Lecture23.pdf

## Physics 220 with Cayon at Purdue University *

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PHYSICS 220 Lecture 23 Kinetic Theory and Thermal Expansion and Textbook Sections 14.4 – 15.5 Lecture 23 Purdue University, Physics 220 1 Overview • Last Lecture – Temperature measure of average Kinetic Energy of molecules – Gas made up of molecules – Ideal Gas Law PV = n R T 2 • P = pressure in N/m (or Pascals) • V = volume in m 3 • n = # moles moles • R = 8.31 J/ (K mole) • T = Temperature (K) • Today – Kinetic Theory of the ideal gas Lecture 23 Purdue University, Physics 220 2 of the – Thermal Expansion Quiz 1) In terms of the ideal gas law, explain briefly how a hot air balloon works . A) Pressure effect B) D i ff ens ty e ect C) Volume effect 2) Which of the following statements is wrong for an of the is an ideal gas: A) P T = n k V B) P V = N k T C) P V R T Lecture 23 Purdue University, Physics 220 3 = n The Ideal Gas Law • PV = Nk B T • Alternate way to write this – N = number of moles (n) x N A molecules/mole – PV= Nk B T •nN A k B T • n(N k )T A B •nRT PV RT• P V = n R T – n = number of moles – R = ideal gas constant = N A k B = 8.31 J/mol/K – R = 1.99 cal/mol/K Lecture 23 Purdue University, Physics 220 4 Ideal Gas Lecture 23 Purdue University, Physics 220 5 Kinetic Theory The relationship between energy and temperature (for monatomic ideal gas) L ()2 fi Δ x xx x x x p p p mv mv mv=−=+−− = • Let N be the number of be the molecules hitting the wall in time Δt • Let L be v Δt N = 1 2 ρV = 1 2 ρAL = 1 2 ρAv x Δt be x • Half of molecules are going away from wall Lecture 23 Purdue University, Physics 220 6 Kinetic Theory The relationship between energy and temperature (for monatomic ideal gas) L 2pmvΔ = 2 x x x L t v Δ= 2 x x avg p mv F tL Δ == Δ 2 N PKE For N molecules, multiply by N 2 3 tr V = x NmvF P A V == 3 2 tr B KEkT= Lecture 23 Purdue University, Physics 220 7 Note = ½m = 3/2m Kinetic Theory 2 13 22 B KEmv kT= = Per molecule 3kT 2 B rms vv m == 33 Internal energy 22 B UNKE NkT nRT=== Lecture 23 Purdue University, Physics 220 8 Example • What is the rms speed of a nitrogen N 2 molecule in this classroom? 3 B KE k T= v = 510 m/s 2 2 13 mv kT = 1150 mph! 22 B = 3kT 2 B v m = 23 2 -27 3(1.38 10 J/K)(273 20)K (28 u)(1.66 10 kg/u) v − ×+ = × × Lecture 23 Purdue University, Physics 220 9 ()( ILQ Suppose you want the rms (root-mean-square) speed of molecules in a sample of gas to double By what factor should in sample of . By what you increase the temperature of the gas? A) 2 B) C) 4 KE = 1 2 mv 2 = 3 2 k B T 2 If d bl 2 dl• If v doubles, v quadruples • Therefore, T quadruples Lecture 23 Purdue University, Physics 220 10 Maxwell-Boltzmann Distribution How many molecules have speeds in a certain range? Lecture 23 Purdue University, Physics 220 11 Collisions between Gas Molecules 22 1 Ad 4 A r d V A π π= = Λ 1 N = ∴Λ= () 2 1 / 4 dNVπ Mf thMean free path: 2 1 2(/)dNVπ Λ= Lecture 23 Purdue University, Physics 220 12 Diffusion 2 rms xDt= D … Diffusion constant Lecture 23 Purdue University, Physics 220 13 Thermal Expansion • Temperature changes can affect many properties of the system • The size of the system will usually change with ff ll d hl itemperature, an e ect ca e thermal expans on • Two specific types of thermal expansion are – Linear expansion – Volume expansion Lecture 23 14Purdue University, Physics 220 Thermal Expansion – Linear • If a piece of metal undergoes a temperature change of ∆T, then its length changes by ∆L α Δ =Δ L T L • α is the coefficient of linear expansion o expansion Lecture 23 Purdue University, Physics 220 15 Thermal Expansion – Volume • The length and height of the solid will also change • If an object undergoes a jg temperature change of ∆T, then its volume changes by ∆Vgy β Δ =Δ V T V • β is the coefficient of volume i o expans on • Fluids will also undergo volume expansions Lecture 23 Purdue University, Physics 220 16 Thermal Expansion • When temperature rises – molecules have more kinetic energy more kinetic • they are moving faster, on the average – consequently, things tend to expand things • Amount of expansion depends on… change in temperature Temp: T – – original length coefficient of thermal expansion Temp: T+ΔT L 0 – expansion •L 0 + ΔL = L 0 + α L 0 ΔT • ΔL = α L 0 ΔT (linear expansion) ΔL • ΔA = 2α A 0 ΔT (area expansion) • ΔV = β V 0 ΔT (volume expansion) Lecture 23 Purdue University, Physics 220 17 Expansion Coefficients L Tα Δ =Δ 0 L Lecture 23 Purdue University, Physics 220 18 Question As you heat a block of aluminum from 0 C to 100 C its density A) Increases B) Decreases C) Stays the same T = 0 C T = 100 C M, V 0 M, V 100 ρ 0 = M / V 0 ρ 100 = M / V 100 < ρ 0 Lecture 23 Purdue University, Physics 220 19 Differential Expansion A bimetallic strip is made with aluminum α=16x10 -6 /K on the left and iron =12x10 -6 /K on the right At left, and α right. room temperature, the lengths of metal are equal. If you heat the strips up what will it look like? strips up, what A B C Al i t l f it tid Lecture 23 Purdue University, Physics 220 20 Alum num gets longer, orces curve so its on outside Thermal Expansion Lecture 23 Purdue University, Physics 220 21 Thermal Expansion of Water • At temperatures well above freezing, water undergoes volume expansion in the usual way • Just above freezing, however, the density of water has a maximum – It cannot be described by the volume expansion equation • Water is very unusual in that it has a maximum density at 4 degrees at C. That is why ice floats, and we exist Lecture 23 Purdue University, Physics 220 22 Water, cont. • The density of ice is also less than the density of water – Water contracts when it melts • Most substances do not behave this way • This unusual behavior of water has many important consequences: – In winter the surface of a lake freezes first – Fish can live in the water underneath and have access to food Lecture 23 23Purdue University, Physics 220 Question Not being a great athlete, and having lots of money to spend, Gill Bates decides to keep the lake in his back yard at the exact temperature which will maximize the buoyant force on him when he swims. Which of the following would be the best choice? A) 0 C 999 90 999.95 1000.00 ) B) 4 C C) 32 C D) 100 C 999.65 999.70 999.75 999.80 999.85 . Density E) 212 C The answer is 4 C, because water has 999.55 999.60 0246810 F B = ρVg its greatest density at 4 C. Since bouyant force is equal to the weight of the displaced fluid or density*volume, when the density is the largest, the bouyant force is maximized. Lecture 23 Purdue University, Physics 220 24 Tight Fit An aluminum plate has a circular hole cut in it. An aluminum ball (solid sphere) has exactly the same diameter as the hole (p when both are at room temperature, and hence can just barely be pushed through it. If both the plate and the ball are now heated up to a few hundred degrees Celsius, how will the ball and the hole fit? A) The ball won’t fit through the hole any more won fit B) The ball will fit more easily through the hole C) Same as at room temperature The ball gets larger but so does the hole! , Since they have the same expansion rate, everything will stay the same! Lecture 23 Purdue University, Physics 220 25 Tight Fit Why does the hole get bigger when the plate expands? Imagine a plate made from 9 smaller pieces. Each piece expands expands. If you remove one piece, it will leave an “expanded hole” Object at temp T Same object at higer T: Pl t d h l bth t l Lecture 23 Purdue University, Physics 220 26 a e and ho e both get larger Stuck Lid ILQ A glass jar (α = 3x10 -6 K -1 ) has a metal lid (α = 16x10 -6 K -1 ) hi h i t k If h t th b l i th i h t tw c s s uc . you ea em y p ac ng them n o wa er, the lid will be A) Easier to open B) Harder to open C) S ame Copper lid expands more, making a looser fit, and easier to open! Lecture 23 Purdue University, Physics 220 27 Jar ILQ A cylindrical glass container (β = 28x10 -6 k -1 ) is filled tthbi ith t ( 208 10 6 k 1 )Ifthto the br m with wa er β = x - - ). If the cup and water are heated 50C what will happen? A) Some water overflows B) Same C) Water below rim Water expands more than Wp container, so it overflows. Lecture 23 Purdue University, Physics 220 28 Summary of Concepts • Kinetic Theory of Monatomic Ideal Gas K 3/2 k T– < tr > = B • Thermal Expansion – ΔL = α L 0 ΔT (linear expansion) (p – ΔV = β L 0 ΔT (volume expansion) Lecture 23 Purdue University, Physics 220 29 Administrator Microsoft PowerPoint - Lecture23 [Compatibility Mode]

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